9514 1404 393
Answer:
(x, y, z) = (-1, 3, 6)
Step-by-step explanation:
The augmented matrix for the system is ...
[tex]\left[\begin{array}{ccc|c}4&-4&4&8\\9&3&1&6\\16&4&1&2\end{array}\right][/tex]
Your graphing or scientific calculator can tell you the solution to this system is ...
(x, y, z) = (-1, 3, 6)
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If you want to solve this by hand, it can work well to divide the first equation by 4 to get ...
x -y +z = 2
This can be subtracted from the other two equations to eliminate z.
(9x +3y +z) -(x -y +z) = (6) -(2) ⇒ 8x +4y = 4
(16x +4y +z) -(x -y +z) = (2) -(2) ⇒ 15x +5y = 0
These two equations can be reduced to standard form:
Subtracting the first equation from the second, we have ...
(3x +y) -(2x +y) = (0) -(1) ⇒ x = -1
Substituting into the first gives y:
2(-1) +y = 1
y = 3 . . . . . . . add 2
Then we can find z from the reduced first equation above:
z = 2 -x +y = 2 -(-1) +3 = 6
Then the solution is (x, y, z) = (-1, 3, 6).
